I get nervous when I see long-time blog readers in my workshops on mathematical modeling with three-act tasks. I tend to assume they’ll be bored. I assume that the pedagogy around these tasks has been self-evident or overly blogged-about these last few years. I should know better. It’s one thing to read about these kinds of tasks. It’s another to do one as a student. After a Saskatoon session last week, for instance, Nat Banting said that the process seemed tighter, and more engineered than he assumed from reading about it.

@dcox21 Very precise implementation. I think teachers view it as "wing it" an open ended. Very calculated.

More than a few people have approached me with the impression that you simply show a photo or a video and then pursue student questions in any direction they take you. Sean Geraghty just asked me to script one of these tasks out with every question I’d ask. I’ll seize that opportunity to post some video of a session I facilitated with teachers this winter around Penny Pyramid in Cambridge and clarify what I think are the important teacher moves in a three-act math task, starting today with act one.

Act One

[00:43] “Here it is. First, I just want you to watch this very brief video.”

[01:27] “Would you go ahead and write down the first question that comes to your mind, if any? No question? That’s perfectly fine.”

[01:45] “Would you introduce yourself to your neighbor and share your question? See if it’s the same question, or a different question.”

[02:28] “I’m really curious what questions are out there. Just toss one out. Who else finds that question interesting?”

[03:04] “I like that you coined a vocabulary term there for us. ‘Layers.'”

[04:24] “I would love to get to all these questions but given limited time we’ll start with these ones up here.”

[04:43] “I want you to write down on a piece of paper your best, gut-level guess for how many coins there are. I’m curious who can guess the closest.”

[05:32] “Would you also write down a number you know is too high – there couldn’t possibly be that many pennies – and a number you know is too low – there couldn’t possibly be that few pennies. Share them with your neighbor.”

Act one attempts to lower barriers to entry. It’s visual. It requires very little literacy from the student. (Notice that I’m using very little formal mathematical vocabulary.) It’s perplexing.

Now look at the student tasks. Students are asked to to watch a video. Students are asked to pose a question. (But if you don’t have one, that’s okay!) Students are asked to decide if they find someone else’s question interesting. Students are asked to guess at a correct answer. Students are asked to decide what an incorrect answer would look like. No one is throwing a hand up saying, “I don’t know where to start.” I don’t know how to make it easier to start a modeling task than this.

I make three promises during act one.

I tell students I’m very curious who guessed closest to the answer.

I tell students I hope we’ll get around to answering all the questions on their list.

I ask students to set an error check on their answer.

I’ll need to make good on each of those promises by the end of act three.

I ask for student questions, but that doesn’t mean you have to. (You don’t have to do any of this of course.)

I have two competing goals in my head in act one. One, I want students to answer the question, “How many pennies are there?” Two, I want to know what questions students have when they see that stupid-huge pile of pennies.

I want to know their questions because students are interesting creatures and, while they spend a lot of time answering questions, they don’t get a lot of opportunities to pose their own. Asking for student questions orients our community around curiosity as a shared value.

But those goals are in conflict. How do you ask students for their questions while knowing, in the back of your head, the question you’re going to pursue. I know some teachers will ask for student questions and then “wait for” or “nudge students towards” the question they want to ask. I suspect this drives students crazy. It drives me crazy, this sense that there’s some question the teacher wants me to ask even while she’s insincerely asking me for my questions.

The quick way around this is to say, “Great. Love these questions. I hope we get to all of them. Here’s one I’ll need your help with first.”

Your Analysis

What did you see in that clip that I didn’t talk about here? What was missing? What would you add? What would you have done differently? Go ahead and constrain yourself to the first act of the task. We’ll pick up tomorrow where I say, “What information do you need here?”

2013 May 9. As usual, a pile of great follow-ups in the comments. Kate Nowak points out a few details that I missed in my discussion. James Cleveland suggests asking for a high and low range before the more precise guess. Great call! Lots of commenters struggle to balance asking for student questions with their curriculum objectives and I respond. So does Math Forum Max. Elaine Watson maps this task to the Standards of Mathematical Practice.

One thing I do when I ask students to guess some of the given information (like the fact that each stack is 13 pennies) is to have each student write their guess on the whiteboard and then have everyone simultaneously show one student, “Bryan.” Then Bryan is tries to ball-park an average of the numbers everyone showed him. It takes about 45 s., but they seem to enjoy the process.

61 Comments

Yeah! Thx for doing this blog series. I thought I was the only one who wasn’t exactly sure how to handle these 3act videos. I have tried one, when I asked for questions (from a room full of 16 year olds), I get endless Qs like “why would anyone build a pyramid like this”. Eventually 1 or 2 quiet kids will offer up a suitably mathematical question.
Looking forward to the rest!

I wonder if when we first introduce kids to these three-act problems, it might help to model responses to these open-ended questions; you can show them, by example, that we are looking for questions with answers that are (in a really broad sense that would include booleans) quantifiable.

Nooope – I think after a teacher encounters you, one of the following happens:
1. They dismiss the idea as too weird and unfamiliar. These people might be beyond saving, but could be swayed if they see kids/participants excited and engaged in meaningful stuff and learning.
2. Too scared to try because they don’t know what it looks like in a classroom
3. Try it but execute badly because they don’t know what it looks like in a classroom, lesson fails, they give up or go though a trial and error process trying to make it work
4. It aligns with how they already conduct lessons anyway and they absorb all your great ideas and beautiful materials into their arsenal of things they’ve collected already.

Video of instruction will go a long way enabling the #2 people to give it a try, and put the #3 people in a position to go kill it much more quickly. And maaaybe sway some #1 people. And give the #4 people ideas about moves to try and tweak. Basically, there is only upside.

So not that you need it, but this is me validating that this is so, so valuable. The potential to affect behavior and effect change results from ways of communicating that live on a spectrum that goes from talking about a thing, to showing a thing, to experiencing a thing. I implied (possibly wrongly) misgivings from your parenthetical “(You don’t have to do any of this of course.)” and want to say: stop that, this is awesome.

Feedback:
add to the 00:43 “I’m curious about what questions this brings about for you.” That’s important – an instruction, even though very general, about what they’re supposed to be doing while watching. Doing something like this with kids who aren’t used to it — it’s important to set an expectation about how we’re learning something here, we’re not screwing around watching youtube.

The next three timestamped elements are think-pair-share. Just an observation, thought it was worth acknowledging — this is what TSP is for and what it looks like in context, done with a purpose.

At 4:43, it’s important that you played the video again.

7:23 — I’m curious about the pros/cons of writing names next to lowest highest guess. On one hand, woo, my name’s on the board, I’m famous. On the other hand, I’m famous for being at an extreme, I might feel bad about that if I am 14. Would it be equally effective to just write the numbers? with the attitude “as a class our estimate is somewhere in here” giving it a more “we’re all in this together” feeling instead of a “who was more right and who was way, way wrong” feeling. Maybe, maybe not, maybe it depends on how supportive/antagonistic the dynamic in your class is, but curious what people think.

Other notes:
I like the choices you made for where to edit out chunks where there is just crowd noise going on. You didn’t edit out all of it, and I appreciate that I still got a sense of the pace.

One of the things that interested me was that you asked for the gut instinct guess first, and then asked for the too high and too low. I typically ask for the latter first, to act as bounds on their actual guess. Is there a reason you do it that way?

Thanks for posting this, I appreciate it. I do have one question/clarification. Because of the break in the video, I wasn’t sure if you asked students to share out their too high/too low guesses. I’ve always had students share them out, but I’ve struggled to get the discourse to be as meaningful as I’ve hoped. I feel like the gut-level estimations of my students have improved dramatically, but I haven’t quite been able to reign in the too high/low responses to be, for example, too low but not ridiculously such (like 3).
I could probably have asked that question much more succinctly: what else do you do with the too high/low guesses? (I hope you haven’t already answered this in the rest of the video…I haven’t watched it because I don’t want to cheat).
Thanks again, looking forward to tomorrow’s post.

M. Hampton

I’ve done penny pyramid with three different classes, and I feel that this particular task has a pretty obvious question (how many pennies?). Most students had this question or some variation of it (how much is it worth? how much does it weight?). I try to pick a question that has some degree of popularity. If most students want to find the weight, I go with that, because I know that it will be just as challenging as my desired question. The math involved will be more or less the same. But what do you do if none of the students come up with your desired question? I hate it when this happens, and I hate having to say, “Hey, those are all great questions…but what I’m really interested in is this.” I think that can be very deflating for students.

I was never a fan of the high/low questions. I always felt like a guess was enough. I suppose it helps with estimation skills, but I feel like I’m already getting that with the guess. Am I missing something here?

To address Kate’s concerns about the guesses, I too never felt comfortable assigning a name to a guess. I also feel bad when a student provides a really bad guess to the class. If just one student criticizes him or her, they’ll never do that again.

I prefer having students write their guesses down on a piece of paper, and I collect that info as I monitor their progress. Later we can look at the data through a box-and-whisker plot or stem-and-leaf or histogram or whatever (no pictographs).

Kate: I would add a bit to your #3: “In addition, feel like they know what it looks like when it’s good, but don’t feel confident they can actually do that.” Think that applies to me a lot – I love the idea, I fear my inability to implement it.

I would also add a #5 – Love the math, love the activity, think it’s really valuable, but not sure how to fit it into the constraints of curriculum, time, admin/parent expectations, etc. etc. Yes, this is the “yeah, but” response, but I think it’s still a real one.

I tend to agree as well about not putting the names next to the guesses – curious as to Dan’s thoughts on that.

James: I had the same question, my gut was to ask for the bounds and then their guess.

Jeff’s question about students guesses that are way off is very similar to one I have for Dan, which is something along the lines of, “The discussion with students looks very different than this discussion with interested, motivated adults. While my students would find this question more interesting than much of what we do, I’m still not sure they would find it interesting enough to really take it seriously.” Maybe this is still my fears of inadequacy of being able to implement this well, and perhaps you are going to address this when you get to the “Why do this?” question your “students’ asked. (I, too, did not watch the rest of the video, waiting for your next post.)

Dan, I really like how you handle getting to the question(s) you want to address, but still honoring their other questions. But what happens if they don’t ask any of the questions you think are important (mathematically) to address? Then we’re back at “here’s some question the teacher wants me to ask even while she’s insincerely asking me for my questions” – which I feel is often where I end up.

Thanks (as always) for sharing this. Looking forward to the next posts.

Zach Brady

Nathan: In regards to getting your students to ask the desired question, I will suggest the following –

Sticking with the popular student questions is a great way to keep them ‘hooked’ (as is mentioned in the video) on the activity. When students pose “How much does it weight/How much is it work”, stay focused on that answer. You can then pose a question to them “Okay, great question! Lets investigate this. It might help to know how much 1 penny weighs/is worth.” You can then provide them with that information or have a student look it up on a computer/tablet/smart phone and then pose “Okay, we know how one penny is worth/weighs, what do we need to know next?” I would hope that at least one student will say that the total number of pennies is needed. At that point you have your original desired question, while still being tied back to the popular question posed by students.

tl;dr: If students are focused on question other than your desired question, perhaps provide them with information that will lead them towards your desired question.

If anyone else has suggestions/critiques, do share!

Mark Watkins

Funny side note… I lured my English teacher wife (who is not on speaking terms with math) into watching the stacking video at the beginning. She stood over my shoulder and watched all the way until you presented the total number of pennies.

Yaacov

My question is when do you introduce the first act, before students have the skills to solve it or after? ie. Do you introduce the first act, then spend a couple days building the skills needed to solve, then reshow Act 1 and let them at it? Or do you build skills first and then go through all the acts in a single class?

Russell Helmstedter

I am inspired by the math problems you present. I have looked through many of your 3-act math problems and have had struggled with how to implement it. This video has cleared up a few things, thank you.

How long does it take the you to work through the entire problem? Say a 56min period? Or does this take more of a 90min period? Yaacov’s question is important as well. When/how do you introduce these problems?

Isaac

Mark, I too am an English teacher. I opened the video on my phone while eating breakfast, plugged it in to the car speakers on my way to work, and walked through the supermarket listening to it. I’m wondering how to adapt and apply the process to English… Any ideas?

Fantastic stuff Dan! I think Kate has already nailed the ‘pedagogy isn’t necessarily obvious’ thing, so thanks for making it explicit.

I think the question of “what happens if your students don’t ask any of the questions you think are important” is an interesting one. Three comments:

1. I guess at some point you need to make the call whether the question(s) you want answered need answering. If so, you might have to go down the, “What about (teacher question)? Who else finds (teacher question) interesting?”

2. The questions you get will give you a good idea if this context is a ‘hook’ for them. If they all say, “Why would anyone care to do that?”, or no-one else finds (teacher question) interesting, then you can choose pull the pin on that example.

3. If your students ask interesting questions that you don’t have data/answer then that’s still ok. You can entertain the question, but using assumptions rather than data. The assumptions can be evaluated at the end. You can then record this as useful data for the next time you use it.

I also find the ‘guess’ thing interesting. I’m guessing (heh) that the ‘guess’ aspect becomes more useful over time, as students hone their estimation skills? I would also be tempted to make more of the ‘guessing process’ at the end, to evaluate the different strategies used by the class to estimate the upper and lower bounds in order to develop these skills.

My experiences have shown that students can get interested in the discussion, but have NO experience owning the learning in the way that 3-acts lets them. A math problem that could go any of a dozen different directions is such a foreign experience to them that they have no idea how to handle it.

This is an open question to anyone: How do you support students in developing a sense of ownership over their experience?

I’ll see students in September who blink their eyes and look back at me when I ask them “What questions come to mind when you look at this?” because they are waiting for me to state the math problem. This is especially true of the students who struggle. Last year I had a full class of struggling math students and fought all year to try to engage them in these types of discussions ultimately to no avail.

I understand the value of this engagement model, but when it comes to the students who most need it, I haven’t figured out the secret to make it work. Is there a scaffolding mechanism, questioning strategy, or otherwise that can help students in this respect?

Lenny VerMaas

When collecting the estimates I like to have the students put them on a number line. I ask for the highest guess, put marks on the board and have them write their estimate on a sticky note and place it on the number line. This provides a nice visual of the estimates. A couple of extensions: Create a box plot of the data to review concepts of median, quartiles, etc. Have the students explain how located their point on the number line.

James Key

Dan, one thing that strikes me just from the timestamps on your transcript is this: you took over *five full minutes* — aka 300+ seconds — just to do Act One. One positive consequence of this is that students have spent 5 full minutes pondering this task they’re about to do, before they’re even ask to do it! (i.e. to do the math associated with the problem) Contrast this with the “bad teacher” implementation, where the problem is read aloud from a handout, and then students are *immediately* asked, “Who can tell me what to do first here?” Dead silence. Three milli-seconds later, “Boy, you guys are quiet today…”

The takeaway for me: learning takes time. Genuine problem-solving takes time. Good teaching takes time. There is no way around this, nor should there be. Some teachers balk at the amount of time I spend on the problems in my lesson. My philosophy is that it’s better to teach *one problem well,* like with serious depth, than to teach 10 problems superficially. But that philosophy is not generally shared by my peers. So it requires a real paradigm shift for people to become willing to do the kind of thing you’re advocating.

I still think we could do better with the too high/too low question. We are trying to have students define the boundaries of a reasonable solution, but too often we get silly or obvious answers to this question.

When I’ve done a 3 Act lesson in class (or any type of lesson of this nature), I’ve found it helpful to define those boundary points: that spot where everything lower is clearly too low, but everything higher is reasonable; and likewise for the high end.

It becomes a richer discussion of what good answers may look like when we are done.

When I first started doing three-acts, I found that students were generally ok diving right into the problem solving process when the task has a low entry point. For instance, I’m doing Dan’s toothpick problem with 7th graders. Dan’s problem is fairly easy for them to solve. You then build in sequels that gradually make things more difficult. (What if Dan’s toothpicks were doubled? How much would it cost to make that? How long would it take?) I think these are all interesting questions and it wouldn’t take much buy-in for the students. Some kids are going to get this stuff done pretty quickly…that’s where you make it tougher. How large of a structure can we fit in this classroom? How many toothpicks? How much would that cost? How long would it take? What if we worked together?

I’ve had plenty of experiences giving a task that was very difficult for the students. This can be very frustrating as students have no idea how to get started. You try to go from group to group and give a little bit of guidance, but with a large class, you can’t get to everyone. Then some students get off-task and start poking each other with pencils and you start freaking out. That’s when you need to get their attention and talk as a class about the first few steps.

I think the timing issue is interesting. My classes are generally 38 minutes long. It is rare for me to get this done in one period. I could easily spend a week on some tasks. And for some students this is frustrating. And for some, this is the first time in their lives where they were really challenged to problem solve and they see that it can be a pain in the ass. But in the end, it’s worth it and it’s rewarding.

If you’re not convinced that this is the thing to do, try Andrew Stadel’s File Cabinet (http://mr-stadel.blogspot.com/2012/04/file-cabinet.html) and watch your entire class get excited as they watch the third act. I still get goosebumps watching them and feeling the energy in the room.

Thanks for the commentary, everybody. In my workshops, I ask teachers to recall my teacher moves and then critique them. It’s always the most interesting part of the session for me.

On question asking.

You guys have me pretty worried here, actually. People are still suggesting various Jedi mind tricks for constraining student questions. This has never gone well for me in class.

The point of asking for student questions isn’t to get them to ask my question or to get them to ask a quantifiable question or to get them to ask better questions. I’m not trying to advance any instructional goal at all. I’m asking for their questions because I’m interested in their questions. It’s a cultural exercise. The best way to wreck it is to imply there’s some right way to be curious when there isn’t.

It doesn’t matter so much if students don’t ask the question that I need to pursue. I just say, “These are great questions and I hope we get to all of them by the end of today. There’s a question I need your help with, though, and that’s ‘how long will it take the paint to dry on the wall?'” Students don’t mind that but only because I understand that I’ve just promised my students that by the end of the task we’ll revisit their questions and I intend to honor it.

Of course, that question should feel natural once it’s asked. Like, “Oh, I didn’t see it that way. Interesting.” No dog bandanas allowed. but I’m not hoping and praying a student will ask it.

Is that any clearer? There’s huge risk and huge reward in this step. Students can walk away feeling valued and creative here or they can walk away feeling small and stepped on. If I need to clarify, let me know.

One of the things that interested me was that you asked for the gut instinct guess first, and then asked for the too high and too low. I typically ask for the latter first, to act as bounds on their actual guess. Is there a reason you do it that way?

No, your way is much better. I’ll give it a shot next time through.

Nathan Kraft posts some useful variations on the process. And then this:

I was never a fan of the high/low questions. I always felt like a guess was enough. I suppose it helps with estimation skills, but I feel like I’m already getting that with the guess. Am I missing something here?

That payoff comes up in act three.

Karl asks about the difference between students and adults. Andrew asks about students who struggle in math. How do we get them to engage in this process?

They’re both right, of course, that this isn’t a representative classroom in a lot of ways. But I used these techniques with struggling students. More to the point, I used these techniques because my students struggled.

ie. The students couldn’t read but I wanted to know the math they knew. So I used more visuals.

ie. The students were freaked out when math problems dumped all kinds of abstractions on them too early in the process. So I delayed all that.

But it does take time to change their assumptions about learning and the culture of a math class. A teacher told me, “You can’t start these tasks in May after teaching a different way for eight months.”

I’d recommend starting with a project like Andrew Stadel’s Estimation180 first. Just show them something. Ask them a question about it. Then ask for a guess about that question. Poll the guesses. Then show the answer and congratulate the winner.

Make it a bell-ringer or a class-ender. I don’t know. But that’s one scaffold towards the larger process.

Then maybe ask them, “What questions they have?” first. Make a quick poll like I did in the clip. But if they get the sense that you have a “right” question in mind here, it’ll set that cultural transformation back.

Sam Olderbak:

Would it be worthwhile for students to justify their guesses?

There is a lot to like about that idea, of course. I don’t do that because I want the problem to be as intuitive and accessible as possible at first. Students like to guess because anybody in class can do it and nobody in class has more access than anybody else to the right answer. Asking for justification loads that task with more expectation than I want to that early in the process.

Yaacov:

My question is when do you introduce the first act, before students have the skills to solve it or after? ie. Do you introduce the first act, then spend a couple days building the skills needed to solve, then reshow Act 1 and let them at it? Or do you build skills first and then go through all the acts in a single class?

I prefer the first model because it gives students a reason to care about the skills we’re developing. Whereas in the second model, the students (presumably) are learning the skills because that’s what’s on the agenda for that day. I want students to have a need for those skills. A general pattern (one which you’ll see later) is: get them good and perplexed by act one, let them work a little bit, let them get stuck, then come in with some instruction.

James, it’s closer to ten minutes, actually, but my hosts edited the clip down. I tell teachers in our debrief that those ten minutes are an investment. They’re costly. The downside is the loss of ten minutes. The upside is students who are more interested in the task and who understand the context better. It’s up to them to decide if the upside defeats the downside.

Russell Helmstedter:

How long does it take the you to work through the entire problem? Say a 56min period? Or does this take more of a 90min period? Yaacov’s question is important as well. When/how do you introduce these problems?

Different problems take longer than others. This would be a full 90-minute task. Others take thirty minutes. And there are ways to compress the process. With this group, I’ve chosen a lengthier problem that highlights more of the interesting things that can happen in a three-act task.

Chris Friberg

Thanks so much for this guidance! It was great to be a student as we did 3-Act activity with you when you came to Reading, MA. I can’t wait to try this one. The challenge for me is definitely in choosing the right questions to ask and when to ask them, without giving away something that they can discover. I think I’m getting better at it, thanks to your modeling.

Dan
What stands out the most to me in terms of the difference between how you implement this task and how I implement them is how you encourage the participants to ask their own questions, but them guide them towards the question you want. You mentioned, “The quick way around this is to say, ‘Great. Love these questions. I hope we get to all of them. Here’s one I’ll need your help with first.'” How often do you actually get around to addressing their question? It seems like more questions are asked than can possibly be answered so students might just think you are brushing them aside.

Nathan
The main reason I like the low and high guesses is to help students check for reasonableness. They often don’t realize when their answers are way to high or low. It also goes well with MP5 that states “They detect possible errors by strategically using estimation and other mathematical knowledge.”

What stands out the most to me in terms of the difference between how you implement this task and how I implement them is how you encourage the participants to ask their own questions, but then guide them towards the question you want.

Ack! No! No guiding! Students hate that. And it’ll result in dead silence the next time I ask them what questions interest them. I don’t guide them towards the question I want. I just ask it. I compliment their questions (sincerely!). I tell them I hope we’ll get around to all of them by the end of the day. That’s also sincere. And then I tell them I need their help with one question in particular.

Sincerity all around! Kids love sincerity!

But, as you mention, this only works if I make a good-faith effort to loop back around to all the questions people asked. Which I do. And it takes like five minutes generally. In the case of the pyramid of pennies, students ask “why?” a lot. That’s easy to talk about briefly. They ask “how heavy?” Easy enough to multiply our answer for “how many?” by the weight of a coin. They ask “how long did that take?” sometimes. I tell them I don’t know. And that’s that.

Kathy Sierra

Re: guessing high/low… there’s evidence that having them *justify* or even explain how they arrived at their guess would change the activity in two ways: 1. if they made the guess intuitively (which we probably assume), they wil not know exactly why they are making that guess — it will be coming from a perceptual awareness they may not be consciously aware of as it is probably not coming from procedural and/or declarative knowledge. 2. If this happens more than a time or two, the students will then *know* that they will need to explain their guess, and this will *change* the guesses they make in the future, and not for the better. (I cringe a little to say this, but the references on the research about explaining guesses can be found in Gladwell’s “Blink”, examples on “rate this jam” and students choosing posters for their room, etc.)

Chris

Bethany

I have used a few of the 3-Act math problems in my Hybrid Intermediate Algebra class (basically Algebra 2) at the Community College where I work. Doing this process with adults is a little different. Yes, they are more interested, but… they also have a lot more preconceived notions about how things should work.
When I them to come up with a question, they often try to make it too ‘mathy.’ They basically try to guess the question that I ‘want’ them to ask, rather than what would really be interesting to them. Also, when I ask them to guess a number… many of my students refuse to guess. They start with trying to calculate the actual number of pennies. So, I have to force a time limit. Even then, many just plain won’t guess if they don’t have time to calculate it out.
Finally, I use personal white boards in my class. I have the students come up with a guess in pairs. I then have them write their guesses and high/low on a white board for each pair. Everyone holds the white boards up at the same time so they can all see everyone’s guesses. Then, at the end of class we compare the calculated values to their guesses to see who came closest.

I’d love to hear suggestions for improving the questioning and guessing process!

Speaking of sincerity and honoring kids’ questions and how to circle around and follow up, here are some things I’ve noticed:

* When we use real problem contexts and can answer kids’ questions about “Why would someone do that?” or “What happened next?” with honest answers, they’re pleased.

* When we use contrived problem contexts and answer kids’ questions with “I’m not sure. Some of these questions we can’t answer without getting Charlie on the phone. Which questions can we answer just using our brains?” and following up on their “why?” or “what next?” questions with “what would you do if you were Charlie?” they are still usually pretty pleased. The task is still meaningful if we’re answering reasonable questions and the kids can imagine what they would do if they were Charlie.

* If students come up with questions that you can’t address in the last few minutes of class, you can hook them for life by being willing to think about their question after school (in that mysterious miasma between school days when students are shocked to learn that you do things like have a life, go to the store, sleep, etc.). Coming back with a response to a kid question you hadn’t anticipated proves your sincere interest in their thinking and about math, showing by your actions that their math ideas are worth thinking about outside of school.

I think you could also do a whole session on understanding who is closer. You’ve measured closer in guesses using the absolute value, but perhaps the ratio of error to size of answer might be more appropriate? Or perhaps, given that many people have a log-based scale of the size of numbers in their head, comparing the ratio of the log of the difference between the numbers may give a different weighting as to who is closest?

I’m also going to model your approach the next time I do a workshop format – start with the concrete ‘what does my teaching approach look like’, have participants ask some questions about it, discuss possible reasons why I use various parts of my approach, and then move into the more abstract ‘why do I use this teaching approach.’

Sean

Just echoing everyone else here, but thank you for this. It’s really great to watch.

Initial impressions:

–It moved a lot faster than I thought it would. From the jump to about 3 minutes in, we have an intro, a video, an opportunity to think on our own, a TPS.
— Love that none of the TPSs linger.
–I liked that you highlighted layer.
–You don’t follow up any of their questions with, “why?” Almost like: oh, glad you find that interesting.

Questions:

–For clarity: why “the first question” and not just “a question?”
–On that same point: Thoughts on “no question? That’s totally fine…” What are the risks of saying, simply, “What questions does this video provoke?” or “What questions do you have about this video?”

Thanks for this series on 3-Acts. I was fortunate to see you facilitate the Penny Pyramid in Palo Alto last summer. However, reading your comments now helps clarify some of the teacher moves that I have forgotten.

Reading the comments from others, I notice that many of their issues are similar to my own issues of facilitating a 3-Act. I have introduced them to teachers over the past year (I’m a math consultant). I’ve always been uncomfortable and a little bit manipulative when I know what question I want them to ask and for which I have the information that they need. I remember you saying about the Penny Pyramid that someone ALWAYS asks how many pennies. In all of times I have done that problem with teachers, it has been one of the questions. I also remember that someone in our Palo Alto class asked, “Why did they build a pyramid”, to which you answered “It was fundraiser” and had a press release about the fundraiser. To other questions that you couldn’t answer, you admitted that you either couldn’t answer it, or that it would be answered after we worked through the problem.

I don’t think that there is such a thing as a “real world problem” in K – 12 education. The students are still using training wheels. In my opinion, the 3-Act process is the closest we are going to get before we remove the training wheels and throw them out into the real world that they will be running when we are, hopefully, collecting Social Security. So we want to train them as well as we can to run a prosperous and productive world, which will, alas, involve non-routine problem solving.

I don’t know if you recall, but after you worked through the Penny Pyramid with our class, I said to you, “The teacher moves you made elicited all 8 of the Common Core Math Practice Standards.” You said, “Well, That’s nice of you to say, but I’m sure I missed some.” I went back to the Palo Alto Quality Inn and wrote up the following summary. I’ve shared this with many teachers as I try to encourage them to include more non-routine problem solving in their classroom. Thanks to you and Andrew Stadel, teachers have a rich treasure trove of engaging problems to access. Hopefully, these problems will encourage teachers to be on the lookout for 3-Act situations of their own.

1. Make sense of problems and persevere in solving them.
• Asked students to brainstorm question “I saw this clip and I had TONS of questions. What was the first question that came to mind?”
• Wrote all questions on board, ranked them, and weighted by interest
• Narrowed down to one question.
• Asked students what information they needed to answer that question. (The teacher had anticipated ahead of time what questions would be asked and what information would be needed.)
• Only gave out information as it was asked for.
• Listened in to groups working and guided them as necessary using questions such as “Have you thought about?” By doing this, the teacher made sure that the students stayed on task and that they were on the right track, but did not use leading or scaffolding questions. The teacher created an environment of trust and safety that encouraged perseverance.
• Noticed that some groups had the answer and then asked an extending question: “How tall would a pyramid be that contained a total of one billion pennies?” • Students brainstormed. They had input into developing questions.
• Once question was decided, students had to determine what information was needed in order to answer the question.
• In order to figure out the information they needed, the students were forced to make sense of the problem.
• When teacher asked, “Have you thought about…”, the students were more comfortable persevering rather than giving up. Also, working within a group helped students to persevere.

2. Reason abstractly and quantitatively.
• Asked students to make a guess…then low and high
• Students reasoned quantitatively about the possible magnitude of the answers in order to develop an estimate. By guessing a high and low estimate, they were forced to reason further about their estimate.

3. Construct viable arguments and critique the reasoning of others.
• Asked students to write down question and share with table (think-pair-share)
• Asked “How many people are wondering that now?” This encouraged students to critique the ideas that others had come up with.
• After writing down their own question, students discussed their question with tablemates, creating the opportunity to construct the argument of why they chose their question, as well as critiquing the questions that others came up with.

4. Model with mathematics.
• The teacher chose an engaging problem that could be modeled mathematically.
• The teacher monitored the students developing the model, but allowed them to develop the model on their own.
• Once the given information was communicated, the students used that information to develop a mathematical model.
• The model involved creating a sum of terms that expressed the value of each layer of the pyramid.

5. Use appropriate tools strategically.
• The teacher monitored the use of tools, but did not interfere.
• In the “debrief”, the teacher discussed the different use of tools. By doing this, students are provided with more tools in their toolbox for future problem solving. Once the model was developed, different tools were used to find the answer:
• Numerical expression
• Summation
• Excel spreadsheet
• BASIC program

6. Attend to precision.
• As students were asking for information about the dimensions of the pyramid, the teacher recorded what they were asking and encouraged them to be very specific in their vocabulary. The teacher, in turn, was very specific about how he labeled the information given.
• Students needed to differentiate between names of the variables that were quantified. For example “stacks” indicated the towers of 13 pennies each, whereas “horizontal layer” indicated the whole layer that was composed of the stacks of 13 pennies. Without this precision,

7. Look for and make use of structure.
• During the “debrief “of the problem, the teacher facilitated a discussion about the structure of the arithmetic expression. What numbers were constant in each term of the sum? Can we use the distributive property? Do we see a pattern?
• The students had to develop an understanding of the physical structure in order to develop a mathematical model that had a numerical structure of its own. The student had to make the connection between the physical structure and the numerical structure of the mathematical model.

8. Look for and express regularity in repeated reasoning. • Early in the discussion of the problem, the teacher helped students develop an understanding of the physical structure of the pyramid. He asked how many stacks of 13 pennies were there on each side of the second lowest horizontal layer. As a discussion ensued, the teacher guided, via strategic questioning and showing strategic images, that there were 39 and not 38.
• Once the base of 40 by 40 was given, the students had to determine how many horizontal layers were involved. There was a regularity of each layer having one fewer set of stacks of 13 pennies, and the top layer had 1 stack. This repeated reasoning allowed them to create their mathematical model.

Many thanks, everybody. For the insightful discussion and pointed commentary. I promoted a bunch of comments to the main post. Great stuff.

Sean, thanks for the mention on “layer.” “Layer” and “stack” seem fairly obvious, I suppose, but I love extending status to as many different kinds of students as possible, incl. those who add words to our classroom lexicon and make it easier to talk about these kinds of contexts.

Let me unpack “What’s the first question that comes to mind, if any?”

I ask for the “first” question because I’m trying to access their rawest, most unmediated perplexity — the first thing they think about before they start wondering, “What does the teacher want me to think about?” I also say “if any” because I want them to know it’s okay if they don’t have any questions about whatever weird thing I happen to have brought along to class that day. That’s okay.

Annie C

I’d like to know more about your integration of vocabulary. You’re right — you use little formal mathematical language at the beginning, and as the “students” use specific terms (layers, stacks), you have them define the terms so the group has a collective vocabulary. I sense that’s a key aspect to lowering the barrier to entry as much as the questioning. No one is asking “Wait, what are we talking about?” there’s no gaming of memorizing random terms out of context, and even the low-literacy student (or ELL) can participate.

I’m a HS ELA and Math teacher, and this, including all the comments, is inspiring. Thank you for sharing, all.

Sean

I ask for the “first” question because I’m trying to access their rawest, most unmediated perplexity — the first thing they think about before they start wondering, “What does the teacher want me to think about?”

Great, thank you.

I’m thinking of “patient problem-solving,” and wondering if there’s an analogy with “patient inquiry.” I watch this and my immediate question was the same as the 61+ folks in that room: how many coins are there?

But then I wondered: how would the questions (and their corresponding popularity) be different if the prompt was “Take a minute and think of a question.” Doubtless you’d have some kids thinking: “What does the teacher want me to think about?” I’m not sure that would be true in your class, though. Or a lot of the teachers in the math blogosphere. And that minute might incent something unusual, something provocative, etc.

I’m interested in the relative advantages that “mediated perplexity” or “patient perplexity” might offer. e.g. If you had a student with an initial question of “how many” that, after a minute, was “not only how many, but let’s convert this thing to euros.”

Sean, I’ve noticed that most of the good questions come out during act one, but once in a while, somebody will ask something else that may be even more interesting during act two. Case in point, when I first did this, I don’t think anyone mentioned the weight of this pyramid. As we were working on it, somebody did wonder how heavy the pyramid was. Personally, I think this is a much more interesting question than “how many?”. That is a huge chunk of solid metal sitting in their living room. How does the table even support all of that weight?

I don’t think you need to limit question asking during act one. In fact, you’ll probably have to pose new questions during act two to keep some of the students challenged…I’m sure Dan will address that in his next post.

Kaci McCoy

Sheila

I started playing with these this year and love them. On Fridays my year10’s have an 80 minute session (60 minutes maths & 20 minutes form time) and are usually counting down for the 20 minute form time to disconnect from the lesson. This week, we did Popcorn Picker and not one of them watched the clock, waiting for the opt out time!
This is wonderful for us to see how it looks when you do it Dan. Thank you sincerely.

Kevin H.

One thing I do when I ask students to guess some of the given information (like the fact that each stack is 13 pennies) is to have each student write their guess on the whiteboard and then have everyone simultaneously show one student, “Bryan.” Then Bryan is tries to ball-park an average of the numbers everyone showed him. It takes about 45 s., but they seem to enjoy the process.

When I started doing 3acts with my pupils (12/13 years old) it struck me that they posed “word problems” such as: “If there are 250k pennies and every penny weights 3 grams, how much does the pyramid weigh?”.

They had so deeply carved that maths was about solving problems with arbitrary numbers that in many occasions they were surprised that the actual number of pennies could be calculated using the actual data!. They expected the pyramid (or the car spiral, or the picture of the lighthouse…) to be the initial point of the problem. They expected a bunch of absurd made-up data and a question at the end…

It took them some time to accept that I was actually interested in their “rawest, most inmediate” question. The basic “how big/tall/large/cool is that?”.

I don’t dare to say that they love these kind of tasks now… but they certainly prefer them to the classical textbook problems.

Katie

Tammy Tidwell

On your spreadsheet of lessons (which I greatly appreciate!!) many of them have the Lesson Plan box checked. I am able to download the videos and use them but I am not able to read or download the lesson plans. I am sure I am missing some important points that could help my students and/or steer them in different directions.

Hi Tammy, thanks for the note. I’m afraid the lesson plan at this point is just the series of questions I include on each page. At some point I may update those with more teacher notes but not in the near future.